Question: Suppose we have a vector-valued function $g(t)$ and a scalar function $f(x, y)$. Let $h(t) = f(g(t))$. We know: $\begin{aligned} &g(5) = (1, 2) \\ \\ &g'(5) = (-3, -3) \\ \\ &\nabla f(1, 2) = (4, 1) \end{aligned}$ Evaluate $\dfrac{d h}{d t}$ at $t = 5$. $h'(5)=$
Formula The multivariable chain rule says that $\dfrac{dh}{dt} = \nabla f(g(t)) \cdot g'(t)$. The $g'(t)$ part is how much a change in $t$ will cause the input to $f$ to move, and the $\nabla f(g(t))$ part is how much $f$ will change in response to this update to its input. [What's the intuition behind the formula?] Applying the formula We want to find $h'(5) = \nabla f(g(5)) \cdot g'(5)$. We know the following. $\begin{aligned} &g(5) = (1, 2) \\ \\ &g'(5) = (-3, -3) \\ \\ &\nabla f(1, 2) = (4, 1) \end{aligned}$ Substituting: $h'(5) = (4, 1) \cdot (-3, -3) = -15$ Answer Therefore, $h'(5) = -15$.